Approximating Directed Weighted-Degree Constrained Networks

• Authors: Zeev Nutov
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

Given a graph H= (V,F) with edge weights {w(e):e茂戮驴 F}, the weighted degreeof a node vin His 茂戮驴 {w(vu):vu茂戮驴 F}. We give bicriteria approximation algorithms for problems that seek to find a minimum cost directedgraph that satisfies both intersecting supermodularconnectivity requirements and weighted degreeconstraints. The input to such problems is a directed graph G= (V,E), edge-costs {c(e):e茂戮驴 E}, edge-weights {w(e):e茂戮驴 E}, an intersecting supermodular set-function fon V, and degree bounds {b(v):v茂戮驴 V}. The goal is to find a minimum cost f-connected subgraph H= (V,F) (namely, at least f(S) edges in Fenter every S茂戮驴 V) of Gwith weighted degrees ≤ b(v). Our algorithm computes a solution of cost , so that the weighted degree of every v茂戮驴 Vis at most: 7 b(v) for arbitrary fand 5 b(v) for a 0,1-valued f; 2b(v) + 4 for arbitrary fand 2b(v) + 2 for a 0,1-valued fin the case of unit weights. Another algorithm computes a solution of cost and weighted degrees ≤ 6 b(v). We obtain similar results when there are both indegree and outdegree constraints, and better results when there are indegree constraints only: a (1,4)-approximation algorithm for arbitrary weights and a polynomial time algorithm for unit weights. Finally, we consider the problem of packing maximum number kof edge-disjoint arborescences so that their union satisfies weighted degree constraints, and give an algorithm that computes a solution of value at least $\lfloor k/36 \rfloor$.